Quadratic Equation Solver

The quadratic formula solves any equation of the form $ax^2 + bx + c = 0$. It states $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. It works for all quadratic equations, even when factoring is impractical.

The discriminant is the expression $b^2 - 4ac$ under the square root. If it is positive, there are two distinct real roots. If it equals zero, there is one repeated real root. If it is negative, there are two complex conjugate roots.

Factoring is fastest when the roots are small integers. Try factoring first by looking for two numbers that multiply to $a \times c$ and add to $b$. If no obvious factors exist, use the quadratic formula directly.

Yes. When the discriminant $b^2 - 4ac$ is negative, the equation has no real solutions. Instead it has two complex conjugate roots of the form $p \pm qi$, where $i = \sqrt{-1}$.

Vieta's formulas relate the roots $r_1$ and $r_2$ to the coefficients without solving: the sum of roots equals $-\frac{b}{a}$ and the product of roots equals $\frac{c}{a}$. These are useful for checking answers or solving problems that ask for the sum or product of roots.